Decay Width and Lifetime

Decay Width and Lifetime

Unstable particles decay with a characteristic mean lifetime $\tau$. Quantum field theory predicts the decay width $\Gamma$, which is the total transition rate summed over all decay modes. They are related by:

$\tau = \frac{\hbar}{\Gamma}$

where $\hbar = 6.582119569 \times 10^{-25}$ GeV·s.

A broader resonance ($\Gamma$ large) lives for a shorter time. The Z boson has $\Gamma_Z = 2.4952$ GeV and lives only $\sim 3 \times 10^{-25}$ s, while the muon has $\Gamma_\mu \sim 3 \times 10^{-19}$ GeV and lives $2.2\,\mu$s.

Exponential Decay

The number of particles surviving at time $t$:

$N(t) = N_0\, e^{-t/\tau}$

Lab Decay Length

A relativistic particle with Lorentz factor $\gamma$ and velocity $\beta c$ travels a mean distance before decaying:

$L_{\text{lab}} = \gamma \beta c \tau$

This is why high-energy pions travel farther before decaying — time dilation extends their laboratory lifetime by $\gamma$.

Your Task

Implement (using $\hbar = 6.582119569 \times 10^{-25}$ GeV·s and $c = 2.998 \times 10^8$ m/s, defined inside each function):

• lifetime_from_width(Gamma_GeV) — returns $\tau = \hbar / \Gamma$ in seconds
• width_from_lifetime(tau_s) — returns $\Gamma = \hbar / \tau$ in GeV
• lab_decay_length(tau_s, gamma, beta) — returns $L = \gamma \beta c \tau$ in metres